How do range and null space of A relate to solving Ax = y?

[squenshl]

Homework Statement

Let x $\in$ R^N, y $\in$ R^M & A $\in$ R^MxN be a matrix. Denote the columns of A by A_k, k = 1,...,N. Let R(A) & N(A) be the range & null space of A respectively.
a) How do the colmuns of A relate to the range of A?
b) Your task is to find the solution to the problem y = Ax, where y & A are known & M = N. What role do R(A) & N(A) play?
c) Let R^N $\ni$ x = (x₁,x₂), x₁ $\in$ R^N₁, x₂ $\in$ R^N₂ & N₁ + N₂ = N. Let A $\in$ R^N₁xN & consider the problem Ax = 0. Assume that you know x₂. Solve for x

Homework Equations

The Attempt at a Solution

a) This is easy, the column space of A is just the range of A.
b) Do we just use the definitions of R(A) and N(A)?
c) I have no idea on this one.
Any help please.

 

[lanedance]

be careful mixing N(A)=Nullspace and N=number of columns - I would use n instead for the number of columns, m for rows

 

[lanedance]

for b), what would happen if y is not in the range of A?

 

[lanedance]

for c) consider writing a as 4 matrices to understand how it works
$$\begin{pmatrix}<br /> B & C \\<br /> D & E<br /> \end{pmatrix}$$

with
B - n1 x n1
C - n1 x n2
D - n2 x n1
E - n2 x n2

then consider the product
$$Ax = \begin{pmatrix}<br /> B & C \\<br /> D & E<br /> \end{pmatrix}x = \begin{pmatrix}<br /> B & C \\<br /> D & E<br /> \end{pmatrix}\begin{pmatrix}<br /> x_1 \\<br /> x_2<br /> \end{pmatrix}$$

bit of an abuse of notation, but hopefully its clear what we're trying to do

 

[squenshl]

Bx₁ + Cx₂ = 0
Dx₁ + Ex₂ = 0

Is that correct mate, if so I'm lost on what to do next?

 

[squenshl]

Still lost on this question.

 

[squenshl]

If x₂ is known what does that mean for x₁ and in turn x.

 

[lanedance]

well solving for x is essentially solving for x_1 as x_2 is known

 

[lanedance]

Now re-arranging the equation we get

$$Ax = \begin{pmatrix} <br /> B & C \\ <br /> D & E <br /> \end{pmatrix}x = \begin{pmatrix} <br /> B & C \\ <br /> D & E <br /> \end{pmatrix}\begin{pmatrix} <br /> x_1 \\ <br /> x_2 <br /> \end{pmatrix} = 0$$

$$Ax = \begin{pmatrix} <br /> B \\ <br /> D <br /> \end{pmatrix}x_1 = - \begin{pmatrix} <br /> C \\ <br /> E <br /> \end{pmatrix} <br /> x_2$$

this is a system of N equations with N1 unknowns